Why? That question was two parts. Need to justify this step; by using the product of the two, you are making the assumption that the two events are independent.1. P(A n B) = the product of the two
You need to use the binomial distribution on this one. You may attempt an answer using the product of the eight individual events, and find it is no different than the probability you found in a or b. This is a wrong assumption though. Because the probability of a 1 or a 0 is 0.5, there is symettry here. Are you familiar with the binomial formula?(c) Find the probability that exactly 1 bit is 1 and 7 bits are 0's.
You need to use Bayes Theorem here. We can say that our events A are of the type of route: congested and not congested. This means our congested can be seen as A1 and A2. These events are EXHAUSTIVE, meaning they cover the entire sample space. However we can have a subevent B which touches are evetns A1 and A2 and is the event that a packet is dropped. Notice that theres no B1 and B2 - just B. Either the packet is dropped or its not dropped. Given that The route is not congested, there is a 0.001 chance of losing a packet; and given that the route is congest, there is a 0.01 chance of a packet. In English: 0.001 of 70% of routes will drop a packet, and 0.01 of 30% of the routes will drop a packet. This is our new sample space: the space of dropped packets. Armed with this, and knowing it follows Bayes, see if you can work the rest of the problem.(a) Find the probability that a packet is dropped.